Skill: rational-expressions

Simplifying Rational Expressions

A rational expression is a fraction with polynomials on top and bottom. Simplify by factoring everything, then canceling common factors — never terms. Every canceled factor leaves an excluded value behind.

1 Understand

The core idea in plain language.

What it is. A rational expression is a fraction with polynomials on top and bottom, like (x²−9)/(x+3). You simplify by factoring everything, then canceling common factors (never terms). Every canceled factor creates an excluded value — an x that would make the original denominator zero. Multiplying and dividing work like fraction arithmetic; adding needs a common denominator.

Why it matters. Rational expressions are where factoring pays off — and they model real rates: combined work, average cost, and anything “per” with a variable in it. They are also the last stop before rational equations.

Where it is used. Combined-work problems · average cost functions · electrical resistance formulas · optics (lens equation).

2 See It

Diagrams that make the idea visual.

Cancel the factor — keep the exclusion

Cancel the common factor (x−2) — but x = 2 is still excluded, because the original denominator cannot be zero. The open circle marks the hole.

Cancel the common factor (x−2) — but x = 2 is still excluded, because the original denominator can’t be zero. The open circle marks the hole.

Terms are not factors

You can only cancel factors — things being multiplied. The 5s here are terms (added/subtracted), so they stay.

3 Worked Examples

Follow each step. The pattern is always the same.

Example 1 Simplify: (x² − 9)/(x + 3)

Factor the top: difference of squares. (x−3)(x+3)/(x+3).
Cancel the common factor (x + 3). = x − 3, x ≠ −3.

(x² − 9)/(x + 3) = x − 3, x ≠ −3

Check: x = 0: (−9)/3 = −3 and 0 − 3 = −3.

Example 2 Simplify: (2x² + 6x)/(4x)

Factor the numerator completely: 2x(x + 3).
Cancel 2x with part of 4x: 2x(x+3)/(4x) = (x+3)/2, x ≠ 0.

(2x² + 6x)/(4x) = (x + 3)/2, x ≠ 0

Check: x = 2: (8+12)/8 = 20/8 = 2.5; (2+3)/2 = 2.5.

Example 3 Multiply: (x/3) · (9/(x−1))

Multiply numerators and denominators: 9x/(3(x−1)).
Reduce 9/3: 3x/(x−1), x ≠ 1.

(x/3) · (9/(x−1)) = 3x/(x−1), x ≠ 1

Check: x = 4: (4/3)·(9/3) = (4/3)·3 = 4; 12/3 = 4.

Example 4 Add: 2/x + 3/(x+1)

Common denominator x(x+1). Rewrite: 2(x+1)/(x(x+1)) + 3x/(x(x+1)).
= (2x + 2 + 3x)/(x(x+1)) = (5x+2)/(x(x+1)), x ≠ 0, −1.

2/x + 3/(x+1) = (5x+2)/(x(x+1)), x ≠ 0, −1

Check: x = 1: 2 + 3/2 = 3.5; (5+2)/2 = 3.5.

4 Common Mistakes

These errors show up on almost every quiz. Spot them now and they will never cost you points.

Mistake 1: Canceling terms instead of factors

Wrong

(x+5)/5 → “x” (canceling the 5s).

Right

(x+5)/5 is already simplest — nothing is being multiplied.

Fix: Ask “are these factors or terms?” Only factors cancel. When in doubt, factor first.

Mistake 2: Forgetting excluded values

Wrong

“(x²−9)/(x+3) = x − 3, done.”

Right

x − 3, x ≠ −3.

Fix: Excluded values come from the original denominator — list them before you cancel anything.

Mistake 3: Adding without a common denominator

Wrong

1/x + 1/y = “1/(x+y)”.

Right

(x+y)/xy.

Fix: Rational expressions add exactly like fractions — common denominator first, always.

5 Quick Check

Try each one on paper first, then reveal the answer.

1. Simplify (x² − 16)/(x − 4). State any excluded values.

Answer

(x−4)(x+4)/(x−4) = x + 4, x ≠ 4
The canceled factor (x − 4) gives the excluded value.

2. Simplify (3x + 12)/(x + 4). State any excluded values.

Answer

3(x+4)/(x+4) = 3, x ≠ −4
Everything cancels — but x = −4 stays excluded.

3. Add: 1/(x−1) + 2/(x+1).

Answer

[(x+1) + 2(x−1)]/((x−1)(x+1)) = (3x−1)/((x−1)(x+1)), x ≠ ±1
Common denominator (x−1)(x+1) first, then add the tops.

Key Points to Remember

Simplify by factoring everything, then canceling common factors — never terms.
Excluded values come from the original denominator — list them before canceling.
A canceled factor becomes a hole; an uncancelled zero stays a vertical asymptote.
Multiply/divide like fractions; add/subtract needs a common denominator.
Dividing also excludes the zeros of the divisor’s numerator.

Practice problems Try the tool See it on FiboMap